Problem 11 - Algebra 2

Question

Complete parts a and b for each quadratic equation. a. Find the value of the discriminant. b. Describe the number and type of roots. Do your answers for Exercises 1, 3, 5, and 7 fit these descriptions, respectively? \(8 x^{2}+18 x-5=0\)

Step-by-Step Solution

Answer

The discriminant is 484, indicating two distinct real, rational roots.

Step 1: Understand the Quadratic Equation

The given quadratic equation is in the form \( ax^2 + bx + c = 0 \) where \( a = 8 \), \( b = 18 \), and \( c = -5 \). Our task is to determine the discriminant and describe the roots.

Step 2: Calculate the Discriminant

The formula for the discriminant \( \Delta \) of a quadratic equation \( ax^2 + bx + c = 0 \) is \( \Delta = b^2 - 4ac \). Here, substitute \( a = 8 \), \( b = 18 \), \( c = -5 \):\[ \Delta = 18^2 - 4 \cdot 8 \cdot (-5) \]\[ \Delta = 324 + 160 = 484 \]

Step 3: Interpret the Discriminant Result

A discriminant \( \Delta \) of 484, which is greater than zero and a perfect square, indicates two distinct real roots. Since it's a perfect square, these roots are rational.

Step 4: Answer Part (b) of the Question

The equation does indeed have two distinct real and rational roots because the discriminant is a positive perfect square (\( \Delta = 484 \)). Now check if the nature of the roots of equations 1, 3, 5, and 7 matches the description given by their discriminants.

Key Concepts

Discriminant

In quadratic equations, the discriminant is a key concept that helps determine the nature of the roots. It is calculated using the formula \( \Delta = b^2 - 4ac \), where \( a \), \( b \), and \( c \) are the coefficients from the quadratic equation \( ax^2 + bx + c = 0 \).

• If the discriminant \( \Delta \) is greater than zero, the quadratic equation has two distinct real roots.
• If \( \Delta \) is equal to zero, the equation has exactly one real root, also known as a repeated or double root.
• If \( \Delta \) is less than zero, there are no real roots, and the solutions are complex or imaginary.

For the example given \(8x^2 + 18x - 5 = 0\), the discriminant is 484 (\(\Delta = 18^2 - 4 \times 8 \times (-5)\)
\(= 324 + 160 = 484\). This indicates there are two real roots. Moreover, because 484 is a perfect square, the roots are rational. Understanding this concept is crucial for solving any quadratic equations as it guides you towards the type of solutions to expect.

Real Roots

Real roots of a quadratic equation are those roots that can be plotted on a real number line. They are the x-values where the quadratic graph touches or crosses the x-axis. Learning about the discriminant helps us identify whether real roots exist:

• When \(\Delta > 0\), there are two distinct real roots, which means the quadratic graph cuts the x-axis at two points.
• When \(\Delta = 0\), there is exactly one real root, indicating the vertex of the parabola just touches the x-axis.

In the sample equation \(8x^2 + 18x - 5 = 0\), we found two real roots because \(\Delta = 484 > 0\). These roots are distinct, meaning the solutions are not repeated. Visualizing these on a graph provides a clear picture of where and how the graph behaves around the x-axis. Real roots provide essential information about the symmetry and vertex of the given quadratic function.

Rational Roots

When talking about rational roots, we refer to the roots of a quadratic equation that can be expressed as a fraction \(\frac{p}{q}\) where both \(p\) and \(q\) are integers, and \(q\) is not zero. Rational roots are more straightforward to handle because they are precise and can be easily calculated.
The nature of rational roots is dictated by the discriminant:

• If the discriminant \(\Delta\) is a perfect square, then the quadratic equation has rational roots.
• This is because the square root of a perfect square discriminant results in an integer value, leading to rational solutions when substituted back into the quadratic formula \(x = \frac{-b \pm \sqrt{\Delta}}{2a}\).

In the example of \(8x^2 + 18x - 5 = 0\), the discriminant 484, being a perfect square, indicates rational roots. Knowing about rational roots helps simplify the solution process and ensures you get exact results without radius of approximation or decimals.